A091149 Expansion of (1 - x - sqrt(1 - 2*x - 23*x^2))/(12*x^2).

1, 1, 7, 19, 109, 421, 2251, 10207, 53593, 263305, 1385263, 7109323, 37728901, 198723565, 1065245299, 5706564247, 30879236017, 167409942289, 913397457367, 4996676997379, 27455383898269, 151263170713909, 836158046041243

Offset: 0

Author

Paul Barry, Dec 22 2003

Keywords

• easy
• nonn

Comments

Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(1,-1), where the U steps come in 6 colors (i.e. Motzkin paths with the up steps in 6 colors), or where the U steps come in 2 colors and the D steps in 3 (or vice versa). Series reversion of x/(1+x+6x^2). - Paul Barry, May 16 2005

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A217275.

Programs

Mathematica

CoefficientList[Series[(1 - x - Sqrt[1 - 2 x - 23 x^2]) / (12 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, May 10 2013 *)

PARI

x='x+O('x^66); Vec((1-x-sqrt(1-2*x-23*x^2))/(12*x^2)) \\ Joerg Arndt, May 11 2013

Formula

G.f.: 2/(1 - x + sqrt(1 - 2*x - 23*x^2)).

a(n) = A014435(n+1)/6.

a(n) = Sum_{k=0..n} binomial(n, k)*6^(k/2)*C(k/2)*(1 + (-1)^k)/2 with C(n)=A000108(n).

a(n) = Sum_{k=0..n} C(n, 2*k)*C(k)*6^k. - Paul Barry, May 16 2005

D-finite with recurrence: (n+2)*a(n) - (2*n+1)*a(n-1) + 23*(1-n)*a(n-2) = 0. - R. J. Mathar, Sep 26 2012

a(n) ~ sqrt(9/4 + 73/(24*sqrt(6)))/(n^(3/2)*sqrt(Pi))*(1 + 2*sqrt(6))^n. - Vaclav Kotesovec, Sep 29 2012

G.f.: 1/(1 - x - 6*x^2/(1 - x - 6*x^2/(1 - x - 6*x^2/(1 - x - 6*x^2/(1 - ....))))), a continued fraction. - Ilya Gutkovskiy, May 26 2017